Why 1 + 2 + 3 + 112 Masoud Khalkhali Mathematics ... · Why 1 + 2 + 3 + = 1 12 Masoud Khalkhali Mathematics Department, UWO, Canada UWO Mathematics Pizza Seminar, Sept 6, 2011

Why 1 + 2 + 3 + · · · = − 112

Masoud Khalkhali

Mathematics Department, UWO, Canada

UWO Mathematics Pizza Seminar, Sept 6, 2011

Abstract

This talk is a quick introduction to summability theory and regularizationof divergent series. Afer a quick introduction to Abel, Cesaro, and Borelsummation techniques, I shall mostly focus on one such theory ofsummability: zeta function regularization. I will show how to define andcompute divergent sums like the one on the title, as well as many otherslike 1 + 1 + 1 + 1 + · · · = −1/2, or 1 + 4 + 9 + 16 + 25 + · · · = 0. I shallalso discuss the closely related cutoff regularization method. One ofEuler’s goals in this area of math was to find (nowadays we say todefine!) the alternating sum of factorials 1!− 2! + 3!− 4! + 5!− · · · . Ishall explain this and will give Euler’s surprising answer!

Introduction

The title of my talk sounds like an utterly wrong statement! After all,the infinite series 1 + 2 + 3 + 4 + · · · is divergent and in fact diverges toinfinity. So shouldn’t we just say 1 + 2 + 3 + 4 + · · · =∞? Of course wecan. But then with the same limited understanding of summation weshall assign the same value, infinite, to a host of other very differenttypes of series like 1 + 1 + 1 + 1 + · · · or 1 + 4 + 9 + 16 + 25 + · · · , etc.The point of my talk is that in doing so we are throwing away a wealth ofinformation hidden in such divergent series. Information that can havepractical implications for mathematics and its applications.

This situation is a bit similar to set theory and cardinal numbers.Mathematicians used to think that there are only two types of numbers:finite and infinite. Of course, after Cantor, we know that there is a vasthierarchy of infinities and knowing about these different types of infinitiesis often very useful.

Enter Euler

One of the first people who realized the importance of divergent seriesand developed some techniques to sum divergent series was LeonhardEuler. In fact Euler was of the opinion that any series is summable andone should just find the right method of summing it! In the last 250years many summation techniques have been designed and there is now avast summability theory: Abel summation, Cezaro summation, Borelsummation, zeta function regularization, cutoff regularization, etc.

The problem: taming infinities

I How to make sense of an infinite sum like

1 + 2 + 3 + · · ·

I Standard answer: it is certainly true that the series is divergent toinfinity and hence we can say

1 + 2 + 3 + · · · =∞

This is correct!

I But imagine we want to regularize this infinity and get a finitenumber. How would you proceed? Somehow we need to extend theclass of convergent series to a larger class. There are manypossibilities.



1 + 2 + 3 + · · ·


1 + 2 + 3 + · · · =∞

This is correct!




1 + 2 + 3 + · · ·


1 + 2 + 3 + · · · =∞

This is correct!


Abel summation

I This is in fact due to Euler. To sum a divergent series like∑

an,define the power series

f (x) =∞∑n=0

anxn.

Assume∑

anxn is convergent for |x | < 1, and assume limx→1− f (x)

exists. Then we can define

∞∑n=1

an := limx→1−

f (x) (Abel).

I Example: The Abel sum of 1− 1 + 1− 1 + · · · is 12 since

f (x) = 1− x + x2 − x3 + · · · = 11+x .

Abelian and Tauberian Theorems

I A Theorem of Abel (this is really due to Abel!) asserts that aconvergent sequence

∑∞n=1 an is Abel summable and its Abel sum is

equal to its standard sum. Thus Abel summation extends thestandard summation of series, but the class of Abel summable seriesis strictly larger than summable series.

I There is a kind of converse to Abel’s theorem. Tauber’s theoremstates that if a series is Abel summable and if an = o(1/n), then theseries is actualy convergent.

Cesaro summation

I Let sk = a1 + · · ·+ ak denote the kth partial sum of the series∑∞n=1 an. The series is called Cesaro summable, with Cesaro sum C ,

if the average value of its partial sums tends to C :

limn→∞

1

n

n∑k=1

sk = C .

I It is easy to see show that a convergent sequence is Cesarosummable and its Cesaro sum is equal to its standard summation.But the converse need not be true:

I Example: the series 1− 1 + 1− · · · is clearly not summable, but isCesaro summable with Cesaro sum 1/2.

Cesaro summation

I Let sk = a1 + · · ·+ ak denote the kth partial sum of the series∑∞n=1 an. The series is called Cesaro summable, with Cesaro sum C ,

if the average value of its partial sums tends to C :

limn→∞

1

n

n∑k=1

sk = C .

I It is easy to see show that a convergent sequence is Cesarosummable and its Cesaro sum is equal to its standard summation.But the converse need not be true:

I Example: the series 1− 1 + 1− · · · is clearly not summable, but isCesaro summable with Cesaro sum 1/2.

I Example: The series 1 + 2 + 3 + · · · is not Cesaro summable as canbe easily verified.

Borel summation

I To sum the divergent series∑

n≥1(−1)nn!, Euler used the formula

n! =

∫ ∞0

e−ttndt

t,

and then wrote, quite formally,∑n≥0

(−1)nn! =∑n≥1

(−1)n∫ ∞

0

e−ttndt

t

=

∫ ∞0

e−t∑n≥1

(−1)ntndt

t=

∫ ∞0

e−t

1 + tdt

Notice that the last integral is convergent!This idea can be extendedto a full fledged summability theory, called Borel summation.

Zeta function regularization

I To regularize a divergent series like∑∞

1 an, we replace it by thefunction

Z (s) =∞∑n=1

a−sn ,

and let∞∑n=1

an := Z (−1).

I For this to make sense: Z (s) must be convergent (henceholomorphic) for Re(s) large enough and it must have an analyticcontinuation to s = −1.

I Let us apply this to the series 1 + 2 + 3 + 4 + · · · .




Z (s) =∞∑n=1

a−sn ,

and let∞∑n=1

an := Z (−1).






Z (s) =∞∑n=1

a−sn ,

and let∞∑n=1

an := Z (−1).



I In this case Z (s) = ζ(s) is the Riemann zeta function, originallydefined as

ζ(s) =∞∑n=1

n−s =1

1s+

1

2s+

1

3s+ · · · R(s) > 1.

It is convergent (and holomorphic) only in R(s) > 1. But it has ananalytic continuation to C \ {1}, with a simple pole at s = 1. So theregularization

1 + 2 + 3 + · · · = ζ(−1)

is well defined and finite. How to compute ζ(−1)? Let us firstunderstand the idea of analytic continuation.

What is analytic continuation and how it is done inpractice?

I Our original formula for ζ(s)

ζ(s) =∞∑n=1

n−s =1

1s+

1

2s+

1

3s+ · · · R(s) > 1,

is divergent in the left half plane R(s) ≤ 1, but ζ(s) has an analyticcontinuation to C \ {1}. How is this possible?

I Standard method to find an analytic continuation of an analyticfunction f (z) : Find a different formula for f (z) which is manifestlydefined and holomorpic on a larger domain.

What is analytic continuation and how it is done inpractice?

I Our original formula for ζ(s)

ζ(s) =∞∑n=1

n−s =1

1s+

1

2s+

1

3s+ · · · R(s) > 1,

is divergent in the left half plane R(s) ≤ 1, but ζ(s) has an analyticcontinuation to C \ {1}. How is this possible?

I Standard method to find an analytic continuation of an analyticfunction f (z) : Find a different formula for f (z) which is manifestlydefined and holomorpic on a larger domain.

Analytic continuation of ζ(s)

I A simple example: f (z) =∑∞

n=0 zn is only convergent and defined

for |z | < 1, but a different formula for it, f (z) = 11−z , is clearly

analytic in C \ {1}.

I We apply the same idea to Riemann zeta function. Is there adifferent formula for ζ(s) that is manifestly analytic in a largerdomain? Yes, and in fact there are many formulas and all of themare rather hard to find. In his 1859 magnificient paper, Riemanngave at least two other formulas for ζ(s) that leads to its analyticcontinuation. Here we give yet another formula that is based onEuler-Maclaurin summation formula. We need to know aboutBernoulli numbers first.

Analytic continuation of ζ(s)

I A simple example: f (z) =∑∞

n=0 zn is only convergent and defined

for |z | < 1, but a different formula for it, f (z) = 11−z , is clearly

analytic in C \ {1}.

I We apply the same idea to Riemann zeta function. Is there adifferent formula for ζ(s) that is manifestly analytic in a largerdomain? Yes, and in fact there are many formulas and all of themare rather hard to find. In his 1859 magnificient paper, Riemanngave at least two other formulas for ζ(s) that leads to its analyticcontinuation. Here we give yet another formula that is based onEuler-Maclaurin summation formula. We need to know aboutBernoulli numbers first.

Enter Bernoulli numbers

I Bernoulli numbers Bm,m = 0, 1, 2, · · · are defined by theirgenerating function:

t

et − 1=∞∑

m=0

Bmtm

m!

I Easy to see that

B0 = 1, B1 = −1

2, B2n+1 = 0, n = 1, 2, 3,

These numbers are ubiquitous: they appear in analysis, geometry,topology, and numerical analysis.

Enter Bernoulli numbers

I Bernoulli numbers Bm,m = 0, 1, 2, · · · are defined by theirgenerating function:

t

et − 1=∞∑

m=0

Bmtm

m!

I Easy to see that

B0 = 1, B1 = −1

2, B2n+1 = 0, n = 1, 2, 3,

These numbers are ubiquitous: they appear in analysis, geometry,topology, and numerical analysis.

Euler-Maclaurin Summation

I This formula turns summation into integration and vice-versa, witha remainder term that can be effectively computed/estimated:

b−1∑k=a

f (k) =

∫ b

a

f (x) dx +n∑

k=1

Bk

k!

(f (k−1)(b)− f (k−1)(a)

)+ Rn.

I A heuristic proof: look for a function g s.t.

g(x + 1)− g(x) = f (x)

Then

f (a)+f (a+1)+· · ·+f (b−1) = g(a+1)−g(a)+· · ·+g(b)−g(b−1)

= g(b)− g(a)


I This formula turns summation into integration and vice-versa, witha remainder term that can be effectively computed/estimated:

b−1∑k=a

f (k) =

∫ b

a

f (x) dx +n∑

k=1

Bk

k!

(f (k−1)(b)− f (k−1)(a)

)+ Rn.

I A heuristic proof: look for a function g s.t.

g(x + 1)− g(x) = f (x)

Then

f (a)+f (a+1)+· · ·+f (b−1) = g(a+1)−g(a)+· · ·+g(b)−g(b−1)

= g(b)− g(a)


I How to find this g? Let D = ddx . Taylor’s formula gives:

f (x) = g(x + 1)− g(x) = (∑ Dn

n!)g(x) = (eD − 1)g(x)

I Rewrite it as

g(x) =D

eD − 1h(x), Dh(x) = f (x), h(x) =

∫ x

a

f (t)dt.

I Solution (Bernoulli numbers appear!)

g(x) = (∞∑n=0

BnDn

n!)h(x).

Notice that g(b)− g(a) = h(b)− h(a) =∫ b

af (x)dx .



f (x) = g(x + 1)− g(x) = (∑ Dn

n!)g(x) = (eD − 1)g(x)

I Rewrite it as

g(x) =D

eD − 1h(x), Dh(x) = f (x), h(x) =

∫ x

a

f (t)dt.


g(x) = (∞∑n=0

BnDn

n!)h(x).


af (x)dx .



f (x) = g(x + 1)− g(x) = (∑ Dn

n!)g(x) = (eD − 1)g(x)

I Rewrite it as

g(x) =D

eD − 1h(x), Dh(x) = f (x), h(x) =

∫ x

a

f (t)dt.


g(x) = (∞∑n=0

BnDn

n!)h(x).


af (x)dx .

Bernoulli polynomials

Bernoulli polynomials Bk(x), k = 0, 1, 2 · · · are defined recursively:

B0(x) = 1

B ′n(x) = nBn−1(x) and

∫ 1

0

Bn(x) dx = 0 for n ≥ 1

Here are the first few

B0(x) = 1

B1(x) = x − 1/2

B2(x) = x2 − x + 1/6

B3(x) = x3 − 3

2x2 +

1

2x


Bernoulli polynomials Bk(x), k = 0, 1, 2 · · · are defined recursively:

B0(x) = 1

B ′n(x) = nBn−1(x) and

∫ 1

0

Bn(x) dx = 0 for n ≥ 1

Here are the first few

B0(x) = 1

B1(x) = x − 1/2

B2(x) = x2 − x + 1/6

B3(x) = x3 − 3

2x2 +

1

2x


I Alternatively, they can be defined by their generating function

text

et − 1=∞∑

m=0

Bm(x)tm

m!.

In particularBm = Bm(0).

I The periodic Bernoulli functions are defined by

B̄n(x) = Bn (x − bxc)


I Alternatively, they can be defined by their generating function

text

et − 1=∞∑

m=0

Bm(x)tm

m!.

In particularBm = Bm(0).

I The periodic Bernoulli functions are defined by


Remainder term for EM summation

I Periodic Bernoulli functions are defined by


Using these, the very important formula for Rn is given by

Rn =(−1)n−1

n!

∫ b

a

f (n)(x)B̄n(x) dx

I Example: for n = 1, we get

b−1∑k=a

f (k) =

∫ b

a

f (x) dx − 1

2(f (b)− f (a))) +

∫ b

a

f ′(x)(x − [x ])dx

Proof: Integration by parts!

Remainder term for EM summation

I Periodic Bernoulli functions are defined by


Using these, the very important formula for Rn is given by

Rn =(−1)n−1

n!

∫ b

a

f (n)(x)B̄n(x) dx


b−1∑k=a

f (k) =

∫ b

a

f (x) dx − 1

2(f (b)− f (a))) +

∫ b

a

f ′(x)(x − [x ])dx

Proof: Integration by parts!

Example: for n = 2, we get

b−1∑k=a

f (k) =

∫ b

a

f (x) dx − 1

2(f (b)− f (a))) +

1

12(f ′(b)− f ′(a)))

−1

2

∫ b

a

f ”(x)B̄2(x)dx

Faulhaber-Bernoulli formulae

I As a first application we get a formula for power sums. Letf (x) = xp. Then the remainder Rn = 0 for n > p and we get anexact formula

n∑k=1

kp =1

p + 1

p∑j=0

(−1)j(p + 1

j

)Bjn

p+1−j .

=1

p + 1np+1 +

1

2np +

p

2B2n

p−1 + · · · .

I Example:

15 + 25 + 35 + · · ·+ n5 =2n6 + 6n5 + 5n4 − n2

12

Faulhaber-Bernoulli formulae

I As a first application we get a formula for power sums. Letf (x) = xp. Then the remainder Rn = 0 for n > p and we get anexact formula

n∑k=1

kp =1

p + 1

p∑j=0

(−1)j(p + 1

j

)Bjn

p+1−j .

=1

p + 1np+1 +

1

2np +

p

2B2n

p−1 + · · · .

I Example:

15 + 25 + 35 + · · ·+ n5 =2n6 + 6n5 + 5n4 − n2

12

A new formula for ζ(s)

I As a second application of Euler-Maclaurin summation we obtain anew formula for ζ(s) that is manifestly extendible to larger domains.Let f (x) = x−s . We get, for s 6= 1,

N∑m=1

1

ms=

1− N1−s

s − 1+

1 + N−s

2

+n∑

k=2

Bks(s + 1) · · · (s + k − 2)(1− N−s−k+1)/k! + Rn

I Let N →∞ with Re(s) > 1 fixed. We get

ζ(s) =1

s − 1+

1

2+

n∑k=2

Bks(s + 1) · · · (s + k − 2)/k!

− 1

n!s(s + 1) · · · (s + n − 1)

∫ ∞1

B̄n(x)x−s−ndx

A new formula for ζ(s)

I As a second application of Euler-Maclaurin summation we obtain anew formula for ζ(s) that is manifestly extendible to larger domains.Let f (x) = x−s . We get, for s 6= 1,

N∑m=1

1

ms=

1− N1−s

s − 1+

1 + N−s

2

+n∑

k=2

Bks(s + 1) · · · (s + k − 2)(1− N−s−k+1)/k! + Rn

I Let N →∞ with Re(s) > 1 fixed. We get

ζ(s) =1

s − 1+

1

2+

n∑k=2

Bks(s + 1) · · · (s + k − 2)/k!

− 1

n!s(s + 1) · · · (s + n − 1)

∫ ∞1

B̄n(x)x−s−ndx


ζ(s) =1

s − 1+

1

2− s

∫ ∞1

(x − [x ]− 1/2)x−s−1dx .

The integral is convergent if Re(s) > 0. This already extends zeta tothe larger domain Re(s) > 0 (with a simple pole at s=1).

I In general, since B̄n(x) is periodic, it is bounded on [1,∞] and hence∫∞1

B̄n(x)x−s−ndx is convergent if Re(s) > 1− n.

I Thus the new expression for ζ(s) shows that it can be analyticallycontinued to the larger domain Re(s) > 1− n. By choosing largerand larger values of n we see that ζ(s) has an analytic continuationto C \ {1}.


ζ(s) =1

s − 1+

1

2− s

∫ ∞1

(x − [x ]− 1/2)x−s−1dx .






ζ(s) =1

s − 1+

1

2− s

∫ ∞1

(x − [x ]− 1/2)x−s−1dx .





Special zeta values

The formula that we obtained before,

ζ(s) =1

s − 1+

1

2− s

∫ ∞1

(x − [x ]− 1/2)x−s−1dx ,

shows that ζ has a simple pole at s = 1 with residue equal to one.

The formula

ζ(s) =1

s − 1+

1

2+

1

12s − 1

2s(s + 1)

∫ ∞1

B̄2(x)x−s−2dx ,

shows that

ζ(0) = −1

2.

Special zeta values

The formula that we obtained before,

ζ(s) =1

s − 1+

1

2− s

∫ ∞1

(x − [x ]− 1/2)x−s−1dx ,

shows that ζ has a simple pole at s = 1 with residue equal to one.

The formula

ζ(s) =1

s − 1+

1

2+

1

12s − 1

2s(s + 1)

∫ ∞1


shows that

ζ(0) = −1

2.

Special zeta values

The formula ,

ζ(s) =1

s − 1+

1

2+

1

12s − 1

2s(s + 1)(s + 2)

∫ ∞1


shows that

ζ(−1) = − 1

12

And in general we get

ζ(−m) = − Bm+1

m + 1, m ≥ 0

Special zeta values

The formula ,

ζ(s) =1

s − 1+

1

2+

1

12s − 1

2s(s + 1)(s + 2)

∫ ∞1


shows that

ζ(−1) = − 1

12

And in general we get

ζ(−m) = − Bm+1

m + 1, m ≥ 0

Special zeta values

These are usually written in a mystifying way:

1 + 1 + 1 + · · · = −1

2

1 + 2 + 3 + · · · = − 1

12

12 + 22 + 32 + · · · = 0

13 + 23 + 33 + · · · =1

120· · · · · ·

1m + 2m + 3m + · · · = − Bm+1

m + 1

These formulas were obtained by Euler in 18th century. His interpretationof the sums were different though.

Cutoff Regularization

I Here is another method to regularize divergent sums.

I Fix a smooth rapidly decreasing function f : [0,∞)→ R, withf (0) = 1. f is called a cutoff or regulator. Replace

∑∞n=0 an by

S(Λ) =∞∑n=0

anf (n

Λ), Λ > 0.

Very often it happens that S(Λ) is convergent, and has anasymptotic expansion near ∞

S(Λ) =∞∑

k=−N

CkΛk , Λ→∞.

Cutoff summation

I In good cases, C0 is independent of the choice of the cutoff functionf . We then define the regularized sum by

a0 + a1 + a2 + · · · = C0.

I Example. If the series is convergent in the usual sense, then C0 isequal to its limit.

I To compute the constant C0, we can try using Euler-Maclaurinsummation formula, as we explain next.

Cutoff summation


a0 + a1 + a2 + · · · = C0.



Cutoff summation


a0 + a1 + a2 + · · · = C0.



Cutoff and Euler-Maclaurin

I In our original Euler-Maclaurin formula, if we leta = 0, b =∞, n =∞, for a rapid decay function g we get anasymptotic expansion

∞∑k=0

g(k) ∼∫ ∞

0

g(x) dx −∞∑k=1

Bk

k!g (k−1)(0).

I For g(x) = xf ( xΛ ), we get an asymptotic expansion

∞∑k=0

kf (k

Λ) ∼ Λ2

∫ ∞0

xf (x) dx − B2

2+ O(

1

Λ).

Cutoff and Euler-Maclaurin

I In our original Euler-Maclaurin formula, if we leta = 0, b =∞, n =∞, for a rapid decay function g we get anasymptotic expansion

∞∑k=0

g(k) ∼∫ ∞

0

g(x) dx −∞∑k=1

Bk

k!g (k−1)(0).

I For g(x) = xf ( xΛ ), we get an asymptotic expansion

∞∑k=0

kf (k

Λ) ∼ Λ2

∫ ∞0

xf (x) dx − B2

2+ O(

1

Λ).


I For g(x) = x2f ( xΛ ), we get an asymptotic expansion

∞∑k=0

k2f (k

Λ) ∼ Λ3

∫ ∞0

xf (x) dx − B3

3+ O(

1

Λ).

I In genral, for g(x) = xmf ( xΛ ), we get an asymptotic expansion

∞∑k=0

kmf (k

Λ) ∼ Λm+1

∫ ∞0

xf (x) dx − Bm+1

m + 1+ O(

1

Λ).


I For g(x) = x2f ( xΛ ), we get an asymptotic expansion

∞∑k=0

k2f (k

Λ) ∼ Λ3

∫ ∞0

xf (x) dx − B3

3+ O(

1

Λ).

I In genral, for g(x) = xmf ( xΛ ), we get an asymptotic expansion

∞∑k=0

kmf (k

Λ) ∼ Λm+1

∫ ∞0

xf (x) dx − Bm+1

m + 1+ O(

1

Λ).

I Thus according to our cutoff regularization scheme, the constantterm should give the regularized sum:

1m + 2m + 3m + · · · = − Bm+1

m + 1

which coincides with our zeta function regularization. This is notaccidental.

I As a direct check, we evaluate, with cutoff f (x) = e−x ,

S(Λ) =∞∑n=1

ne−nΛ =

a

(1− a)2, a = e−

1Λ ,

which has the expansion

S(Λ) = Λ2 − 1

12+

1

Λ2

1

240+ O(

1

Λ4).

I Thus according to our cutoff regularization scheme, the constantterm should give the regularized sum:

1m + 2m + 3m + · · · = − Bm+1

m + 1

which coincides with our zeta function regularization. This is notaccidental.

I As a direct check, we evaluate, with cutoff f (x) = e−x ,

S(Λ) =∞∑n=1

ne−nΛ =

a

(1− a)2, a = e−

1Λ ,

which has the expansion

S(Λ) = Λ2 − 1

12+

1

Λ2

1

240+ O(

1

Λ4).

I This matches perfectly with what we got from Euler-Maclaurinsummation formula.

I As a good excercise, one should check by a direct calculation asabove that for the cutoff f (x) = e−x one obtains the same result for∑∞

k=1 km as we found using Euler-Maclaurin summation formula.

Summary

I We sketched several approaches to regularizing divergent series like1 + 2 + 3 + · · · : Abel summation, Cesaro summation, Borelsummation, zeta function regularization, and cutoff regularization.The zeta regularization needed more sophisticated tools like analyticcontinuation, and computing special zeta values which was doneusing Euler-Maclaurin summation formula.

Why 1 + 2 + 3 + 112 Masoud Khalkhali Mathematics ... · Why 1 + 2 + 3 + = 1 12 Masoud Khalkhali Mathematics Department, UWO, Canada UWO Mathematics Pizza Seminar, Sept 6, 2011

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